Optimizing Nonlinear Problem Solving: Novel Third to Eighth-Order Iterative Schemes via Homotopy Perturbation Method

This paper presents a comparison of the newly proposed third, sixth, seventh, and eighth-order iterative schemes (NM3, NM6, NM7, NM8) with existing methods for solving nonlinear problems. The methods are evaluated against several established schemes, including those by Amat et al. (AM), Kim (KI), Maheshwari (MA), and others. Real life problems are solved using these iterative schemes. The analysis includes a detailed numerical comparison, convergence behavior, and the effectiveness of these methods. Results indicate that proposed methods demonstrate superior performance in terms of convergence speed and accuracy. The newly proposed methods also generate larger basins of attraction, indicating better stability and reliability in solving nonlinear equations. Math classification 2020: 65H20, 90C39